Open Quantum Systems: Density Matrix Formalism and Applications The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Tempel, David G., Joel Yuen-Zhou, and Alan Apuru-Guzik. 2012. Open quantum systems: density matrix formalism and applications. Fundamentals of Time-Dependent Density Functional Theory. Lecture Notes in Physics 837:211-229. Published Version doi:10.1007/978-3-642-23518-4_10 Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:8472055 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Open Access Policy Articles, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#OAP 9 Open quantum systems: Density matrix formalism and applications D. G. Tempel, J. Yuen-Zhou and A. Aspuru-Guzik 9.1 Introduction In its original formulation, TDDFT addresses the isolated dynamics of elec- tronic systems evolving unitarily [Runge 1984]. However, there exist many situations in which the electronic degrees of freedom are not isolated, but must be treated as a subsystem imbedded in a much larger thermal bath. The theory of open quantum systems (OQS) deals with precisely this situation, in which the bath exchanges energy and momentum with the system, but particle number is typically conserved. Several important examples include vibrational relaxation of molecules in liquids and solid impurities, coupling to a photon bath in cavity quantum electrodynamics, photo-absorption of chromophores in a protein environment, electron-phonon coupling in single- molecule transport and exciton and energy transfer nanomaterials. Even with simple system-bath models, describing the reduced dynamics of many corre- lated electrons is computationally intractable. Therefore, applying TDDFT to OQS (OQS-TDDFT) offers a practical approach to the many-body open- systems problem. The formal development of the theory of OQS begins with the full unitary dynamics of the coupled system and bath, described by the Von Neumann equation for the density operator dˆρ(t) = −i[Hˆ (t), ρˆ(t)] . (9.1) dt Here, Hˆ (t)= HˆS(t)+ αHˆSB + HˆB (9.2) is the full Hamiltonian for the coupled system and bath and N N 1 Hˆ (t)= − ∇2 + v (r , r )+ v (r ,t) (9.3) S 2 i ee i j ext i i=1 i<j i X X X is the Hamiltonian of the electronic system of interest in an external potential vext(r,t). This potential generally consists of a static external potential due to the nuclei and an external driving field coupled to the system such as a laser field. For an interacting electronic system, vee(r, r′)=1/|r − r′| is 9 Open quantum systems: Density matrix formalism and applications 183 the two-body Coulomb repulsion. The system-bath coupling, HˆSB, acts in the combined Hilbert space of the system and bath and so it couples the two subsystems. Typically, for a single dissipation channel, the system-bath coupling is taken to have a bilinear form, HˆSB = −Sˆ ⊗ Bˆ , (9.4) where Bˆ is an operator in the bath Hilbert space which generally couples to ˆ N a local one-body operator S = i=1 sˆ(pˆi, rˆi) in the system Hilbert space. Implicit in OQS is a weak interaction between the system and bath, so that hP i one can treat the system-bath coupling perturbatively by introducing the small parameter α as in Eq. (9.2). HˆB is the Hamiltonian of the bath, which will typically consist of a dense set of bosonic modes such as photons or phonons. The density of states of HˆB determines the structure of reservoir correlation functions, whose time-scale in turn determines the reduced system dynamics. The goal of the theory of open quantum systems is to arrive at a reduced description of the dynamics of the electronic system alone, by integrating out the bosonic modes of the bath. In this way, one arrives at the quantum master equation, which describes the non-unitary evolution of the reduced system in the presence of its environment. In the next section, we derive the many- electron quantum master equation and discuss common approximations used to treat the system-bath interactions. We then formulate the master equation approach to TDDFT rigorously, by establishing a van Leeuwen construction for OQS. Next, we turn to a practical Kohn-Sham (KS) scheme for dissipative real-time dynamics and finally discuss the linear response version of OQS- TDDFT, giving access to environmentally broadened absorption spectra. 9.2 The generalized quantum master equation We begin this section by deriving the formally exact many-electron quantum master equation using the Nakagima-Zwanzig projection operator formalism [Nakajima 1958, Zwanzig 1960, Zwanzig 2001]. Using projection operators, the master equation can be systematically derived from first principles start- ing from the microscopic Hamiltonian in Eq. (9.2) (i.e. without phenomeno- logical parameters). This is particularly amenable to TDDFT, in which the electronic degrees of freedom are treated using first principles as well. We will then discuss the Born-Markov approximation and the widely used Lindblad master equation [Lindblad 1976]. 184 D. G. Tempel, J. Yuen-Zhou and A. Aspuru-Guzik 9.2.1 Derivation of the quantum master equation using the Nakagima-Zwanzig projection operator formalism Our starting point is Eq. (9.1) for the evolution of the full density operator of the coupled system and bath, dˆρ(t) = −i[Hˆ (t), ρˆ(t)] ≡−iL˘(t)ˆρ(t) , (9.5) dt where L˘(t) is the Liouvillian superoperator for the full evolution defined by Eq. (9.5). It may be separated into a sum of Liouvillian superoperators as L˘(t)= L˘S(t)+ L˘SB + L˘B , (9.6) where each term acts as a commutator on the density matrix with its respec- tive part of the Hamiltonian. Our goal is to derive an equation of motion for the reduced density operator of the electronic system, ρˆS(t) = TrB{ρˆ(t)} , (9.7) defined by tracing the full density operator over the bath degrees of freedom. To achieve this formally, we introduce the projection superoperators P˘ and Q˘. The operator P˘ is defined by projecting the full density operator onto a product of the system density operator with the equilibrium density operator of the bath, ˘ eq P ρˆ(t)=ˆρB ρˆS(t) . (9.8) Q˘ =1 − P˘ projects on the complement space. In this sense, P˘ projects onto the degrees of freedom of the electronic system we are interested in, while Q˘ projects onto irrelevant degrees of freedom describing the bath dynamics. Using these projection operators, Eq. (9.5) can be written formally as two coupled equations: d P˘ρˆ(t)= −iP˘L˘ρˆ(t)= −iP˘L˘P˘ρˆ(t) − iP˘L˘Q˘ρˆ(t) (9.9a) dt d Q˘ρˆ(t)= −iQ˘L˘ρˆ(t)= −iQ˘L˘P˘ρˆ(t) − iQ˘L˘Q˘ρˆ(t) . (9.9b) dt If Eq. (9.9b) is integrated and substituted into Eq. (9.9a), one obtains d P˘ρˆ(t)= −iP˘L˘P˘ρˆ(t) dt t t ′ ′ t i R dτ Q˘L˘(τ ) i R dτ Q˘L˘(τ) − dτ P˘L˘e− τ Q˘L˘P˘ρˆ(τ) − iP˘L˘e− 0 Q˘ρˆ(0) . (9.10) Z0 By performing a partial trace of both sides of Eq. (9.10) over the bath degrees of freedom, one arrives at the formally exact quantum master equation t dˆρS(t) = −i[Hˆ (t), ρˆ (t)] + dt′K˘ (t, τ)ˆρ (τ)+ Ξ(t) . (9.11) dt S S S Z0 9 Open quantum systems: Density matrix formalism and applications 185 Here, t ′ ˘ ˘ ′ ˘ ˘ ˘ i Rτ dτ QL(τ ) ˘ ˘ eq K(t, τ) = TrB P Le− QLρˆB (9.12) is the memory kernel. n o t i R dτQ˘L˘(τ) Ξ(t) = TrB −iP˘L˘e− 0 Q˘ρˆ(0) (9.13) n o arises from initial correlations between the system and its environment [Meier 1999]. Equation (9.11) is still formally exact, asρ ˆS(t) yields the exact expec- tation value of any operator depending on the electronic degrees of freedom. In practice, approximations to the memory kernel and initial correlation term are needed. 9.2.2 The Markov approximation One often invokes the Markov approximation, in which the memory kernel is local in time and the initial correlations vanish, i.e. t dt′K˘ (t,t′)ˆρS(t′)= D˘ρˆS(t) (9.14) Z0 and Ξ(t)=0 . (9.15) The Markov approximation is valid when τS ≫ τB is satisfied, where τS is the time-scale for the system to relax to thermal equilibrium and τB is the longest correlation time of the bath [Breuer 2002]. Roughly speaking, the memory of the bath is neglected because the bath decorrelates from itself before the system has had a chance to evolve appreciably [Van Kampen 1992]. The time- scale τS is inversely related to the magnitude of the system-bath coupling, and so a weak interaction between the electrons and the environment is implicit in this condition as well. The Lindblad form of the Markovian master equation, 1 1 D˘ρˆ (t)= Sˆρˆ (t)Sˆ† − Sˆ†Sˆρˆ (t) − ρˆ (t)Sˆ†Sˆ , (9.16) S S 2 S 2 S is constructed to guarantee positivity of the density matrix. This is desirable, since the populations of any physically sensible density matrix should remain positive during the evolution. In the Lindblad equation, in addition to the Markov approximation, one performs second-order perturbation theory in the system-bath interaction [Eq. (9.4)]. These two approximations in tandem are referred to collectively as the Born-Markov approximation.